LOS ANGELES , California -LRB- CNN -RRB- -- Natalie Cole 's search for a new kidney ended this week when someone with a compatible organ died and their family asked that it be given to the singer , according to the organ procurement group that handled the donation .

Natalie Cole received a kidney transplant Tuesday in Los Angeles .

Cole , who underwent a successful kidney transplant at Cedars-Sinai Medical Center in Los Angeles Tuesday , was `` responding well and recuperating '' Thursday , her publicist said .

Cole had been on what she called `` a very long list '' of people waiting for an organ donation when she got word recently that OneLegacy , an organ procurement agency in Southern California , had a kidney that was a biological match .

It was a `` directed donation '' from a deceased organ donor , according to OneLegacy CEO Tom Mone .

`` Having heard of Ms. Cole 's need for a kidney , the family asked that one of their loved one 's kidneys be transplanted to Ms. Cole if they were a match , '' Mone said .

Directed organ donations are rare , and rarer still are those directed to someone not related to or known personally by the deceased , he said .

The donor will remain anonymous until the family gives permission for the name to be made public , he said .

Cole 's need for a kidney donation was highly publicized , including with an appearance on CNN 's `` Larry King Live '' on March 31 .

`` I 'm on a very long list , which is why we are looking to donors , '' said Cole , the daughter of legendary crooner Nat King Cole .

Cole said her kidney troubles date to February 2008 when she was diagnosed with Hepatitis C. Cole said she attributes the Hepatitis C to her well-publicized struggles to stop using cocaine and heroin . Cole said she has been sober for some time now after two stints in a rehabilitation clinic .

Cole said she underwent chemotherapy in an aggressive way to fight the virus . Within four months of getting chemotherapy , both of Cole 's kidneys failed .

`` I could n't breathe . I -- I went into -- literally , my kidneys stopped functioning . They stopped , you know , processing the fluid that was starting to build up in my body . ''

Since then Cole has been on dialysis three days a week and has been searching for a kidney , she told King in one of the first public interviews about the issue .

Cole won six Grammys for her 1991 critically acclaimed album `` Unforgettable : With Love , '' a jazzy tribute to her father . She won a Grammy this year for her new album `` Still Unforgettable '' and another Grammy for her work on another album .

Cole was still able to tour this year even with her kidney ailment . She canceled a planned show last week in South Korea because of her illness , which meant she was in Los Angeles when the organ became available .

`` I have been on dialysis in Istanbul , Milan , Indonesia , Manila , London . It 's -- it 's amazing , '' Cole said .

During King 's show , dozens of people sent e-mails to CNN telling Cole that they would get tested to see if their kidney could be donated .

King handed a thick stack of paper to Cole .

`` These are all e-mails from dozens -- dozens of people offering to be tested to see if they can match , who want to give you a kidney , '' King said .

Cole stared at the papers for a moment .

`` There are some great human beings out there . That 's all I can say , '' Cole said .

Cole is expected to return to the tour after three to four months of recovery .

CNN 's Denise Quan contributed to this report .

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Natalie Cole 's new kidney came from deceased fan ; fan 's family asked for Cole

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Natalie Cole underwent kidney transplant surgery Tuesday

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Singer received supportive e-mails when on CNN 's `` Larry King Live '' in March